Subgroups — #24984297

Order 11008.

190 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582725	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063259	—	2	1	no	—
5	C₂×C₄₃⋊C₃₂	—	2	1	no	—
6	C₂×C₄₃⋊C₃₂	—	2	1	no	—
7	C₂×C₄₃⋊C₃₂	C₄	1	2	yes	complemented
8	#14582725	—	2	1	no	—
9	#14582725	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×#107×C₄₃	—	2	1	no	—
12	C₄₃×#366	—	2	1	no	—
13	C₄₃×#366	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₃	—	4	1	no	—
15	#13063259	—	2	2	no	—
16	#13063259	—	4	1	no	—
17	C₄₃⋊C₃₂	—	2	2	no	—
18	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₃	—	4	1	no	—
20	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₃	Q₈	1	1	yes	characteristic
26	#107×C₄₃	—	2	2	no	—
27	#107×C₄₃	—	4	1	no	—
28	C₂²×C₄×C₄₃	—	8	1	no	—
29	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
30	C₂²×Dic₄₃	—	8	1	no	—
31	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₄₃	Q₁₆	1	1	yes	characteristic
37	C₄⋊C₄×C₄₃	—	4	2	no	—
38	C₄⋊C₄×C₄₃	—	8	1	no	—
39	C₄⋊Dic₄₃	—	4	2	no	—
40	C₄⋊Dic₄₃	—	8	1	no	—
41	C₁₆×C₄₃	C₄²	1	1	yes	characteristic
42	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₁₆×C₄₃	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₄₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₄₃	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₄₃	—	8	2	no	—
47	C₂×C₄×C₄₃	—	16	1	no	—
48	C₂×C₄×C₄₃	—	16	1	no	—
49	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₄₃	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₄₃	Q₃₂	1	1	yes	characteristic
55	C₂×Dic₄₃	—	8	2	no	—
56	C₂×Dic₄₃	—	16	1	no	—
57	C₂×Dic₄₃	—	16	1	no	—
58	C₈×C₄₃	#95	1	1	yes	characteristic
59	C₈×C₄₃	#103	1	1	yes	characteristic
60	C₈×C₄₃	#103	1	1	yes	characteristic
61	C₈×C₄₃	#106	1	1	yes	characteristic
62	#7471	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	#357	1	1	yes	characteristic
64	C₂²×C₄₃	#357	1	1	yes	characteristic
65	C₂²×C₄₃	#366	1	1	yes	characteristic
66	C₂²×C₄₃	D₃₂	1	1	yes	characteristic
67	C₂²×C₄₃	SD₆₄	1	1	yes	characteristic
68	C₂²×C₄₃	SD₆₄	1	1	yes	characteristic
69	C₂²×C₄₃	Q₆₄	1	1	yes	characteristic
70	C₄×C₄₃	—	16	2	no	—
71	C₄×C₄₃	#340	1	1	yes	characteristic
72	C₄×C₄₃	#358	1	1	yes	characteristic
73	C₄×C₄₃	#358	1	1	yes	characteristic
74	C₄×C₄₃	#367	1	1	yes	characteristic
75	Dic₄₃	—	16	2	no	—
76	C₂²×C₃₂	—	43	1	no	—
77	C₂×#366	—	43	1	no	—
78	C₂×#366	D₄₃	1	1	yes	characteristic
79	C₂×C₄₃	#1380	1	1	yes	characteristic
80	C₂×C₄₃	#1415	1	1	yes	characteristic
81	C₂×C₄₃	#1415	1	1	yes	characteristic
82	C₂×C₄₃	#1416	1	1	yes	characteristic
83	C₂×C₄₃	#1416	1	1	yes	characteristic
84	C₂×C₄₃	#1423	1	1	yes	characteristic
85	C₂×C₄₃	#1424	1	1	yes	characteristic
86	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	86	1	no	—
89	C₂×C₃₂	—	43	2	no	—
90	C₂×C₃₂	—	86	1	no	—
91	C₂×C₃₂	—	86	1	no	—
92	#366	—	2	1	no	—
93	#366	—	43	2	no	—
94	#366	—	86	1	no	—
95	#366	Dic₄₃	1	2	yes	complemented
96	C₄₃	#7471	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄⋊C₄	—	172	1	no	—
100	C₂×C₁₆	Dic₈₆	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
103	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
106	#107	—	2	2	no	—
107	#107	—	4	1	no	—
108	#107	—	86	2	no	—
109	#107	—	172	1	no	—
110	C₃₂	—	86	2	no	—
111	C₂²×C₄	—	8	1	no	—
112	C₂²×C₄	—	344	1	no	—
113	C₂²×C₄	#10621882	1	1	yes	characteristic
114	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
115	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
116	C₂×C₈	#10621883	1	1	yes	characteristic
117	C₂×C₈	#10621884	1	1	yes	characteristic
118	C₂×C₈	#10621885	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	344	1	no	—
123	C₄⋊C₄	—	172	2	no	—
124	C₁₆	C₄×Dic₄₃	1	1	yes	characteristic
125	C₁₆	Dic₄₃⋊C₄	1	1	yes	characteristic
126	C₁₆	C₄⋊Dic₄₃	1	1	yes	characteristic
127	C₁₆	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
128	C₂³	#13063266	1	1	yes	characteristic
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	344	2	no	—
133	C₂×C₄	—	688	1	no	—
134	C₂×C₄	—	688	1	no	—
135	C₂×C₄	#13063245	1	1	yes	characteristic
136	C₂×C₄	#13063246	1	1	yes	characteristic
137	C₂×C₄	#13063258	1	1	yes	characteristic
138	C₂×C₄	#13063268	1	1	yes	characteristic
139	C₂×C₄	#13063269	1	1	yes	characteristic
140	C₂×C₄	#13063271	1	1	yes	characteristic
141	C₈	C₄₃⋊#95	1	1	yes	characteristic
142	C₈	#13063248	1	1	yes	characteristic
143	C₈	#13063249	1	1	yes	characteristic
144	C₈	#13063256	1	1	yes	characteristic
145	C₂²	#14582699	1	1	yes	characteristic
146	C₂²	#14582700	1	1	yes	characteristic
147	C₂²	#14582724	1	1	yes	characteristic
148	C₂²	#14582734	1	1	yes	characteristic
149	C₂²	#14582736	1	1	yes	characteristic
150	C₂²	#14582737	1	1	yes	characteristic
151	C₂²	#14582739	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	688	2	no	—
154	C₄	#14582661	1	1	yes	characteristic
155	C₄	#14582702	1	1	yes	characteristic
156	C₄	#14582703	1	1	yes	characteristic
157	C₄	#14582726	1	1	yes	characteristic
158	C₂	#19132686	1	1	yes	characteristic
159	C₂	#19132772	1	1	yes	characteristic
160	C₂	#19132773	1	1	yes	characteristic
161	C₂	#19132775	1	1	yes	characteristic
162	C₂	#19132776	1	1	yes	characteristic
163	C₂	#19132794	1	1	yes	characteristic
164	C₂	#19132796	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063266
commutator	C₁₆×C₄₃	C₄²
Frattini	C₂²×C₁₆	C₂×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle, prime radical	C₂³×C₄₃	D₁₆
cosocle	C₂²×C₁₆×C₄₃	C₂²
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#7471
LCS	G ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. C₂×C₄₃⋊C₃₂	C₄	semidirect
9. #14582725	C₄	semidirect
13. C₄₃×#366	C₄	semidirect
95. #366	Dic₄₃	semidirect
96. C₄₃	#7471	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄₃⋊C₃₂	2
C₂×#14582725	2
C₂×C₄₃×#366	2
#7471	43